Below is a list that I have developed for our Defensive & Evasive Driving Course that is both simple and effective for use by protective agents who routinely check their vehicles for optimal readiness, including rental vehicles. 

The acronym TFOLDE  (A fold) is used as a simple reminder of what to do in case a checklist is not in hand:

TFOLDE

Tires:

Air pressure in the tires: Pressure, check tire type (good for vehicle and environment?) and status of wear? (SWISS+TECH Bodygard)

Air Intake System: Filter clean, intake system ok? (connections are good, no damage, etc.)

Fuel:

Full tank? Consider range of full tank (look up vehicles MPG on caranddriver.com)

Oil:

Check oil level and freshness. 

Lights/Visibility:

Are all lights operable and bright enough? Electrical system is ok? Windows clean? RainX?

Damage/Insurance:

Note any damage; is it going to cause an issue? Consider Insurance: standard/liability coverage, CDW/LDW, etc. *Verify insurance covers intended usage and drivers*

Equipment:

MedKit, battery jumper/jumper cables, all device charger attachments, tire puncture kit/inflator, umbrella and disposable rain coats, traffic cones/flares, evac tool (SWISS+TECH Bodygard)

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For Training Schedule and Locations see here